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Question
\[\lim_{x \to 2} \left( \frac{x}{x - 2} - \frac{4}{x^2 - 2x} \right)\]
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Solution
\[\lim_{x \to 2} \left[ \frac{x}{x - 2} - \frac{4}{x^2 - 2x} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{x}{x - 2} - \frac{4}{x\left( x - 2 \right)} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{x^2 - 4}{x\left( x - 2 \right)} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{\left( x - 2 \right)\left( x + 2 \right)}{x\left( x - 2 \right)} \right]\]
\[ = \lim_{x \to 2} \left[ \frac{\left( x + 2 \right)}{x} \right]\]
\[ = \frac{2 + 2}{2}\]
\[ = 2\]
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